Pure Mathematics Seminar (Colloquium)
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Organizer: Pantelis Eleftheriou
Organizer: Pantelis Eleftheriou |
Upcoming talks:Title: Elliptic curves, special points, and unlikely intersections Abstract: This will be a talk aimed at a wide audience. In the first part, I will recall the notion of an elliptic curve over the complex numbers and I will review the beautiful theory that allows us to classify them. Following this, I will recall what it means for an elliptic curve to have complex multiplication (CM) and I will discuss major advances over the last 30-40 years that describe how tuples of CM elliptic curves are distributed. Finally, I will venture into the realm of "unlikely intersections", which aims to study the distribution of other forms of "additional structure" on elliptic curves and their generalizations. Title: TBA Abstract: TBA Past talks:Title: Classifying Bijective Set-theoretic Solutions to the Pentagon Equation Abstract: In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions, including the notion of irretractable solutions, and explore how these solutions correspond with matched pairs of groups. Finally, I will show how each irretractable solution lifts to provide the full classification of all bijective solutions. Title: Recognizing groups in Erdős geometry and model theory Abstract: Erdős-style geometry is concerned with difficult questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be viewed as asking for the possible number of intersections of a given algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups. Techniques from model theory -- variants of Hrushovski's group configuration and Zilber's trichotomy principle -- are very useful in recognizing these groups, and led to far reaching generalizations of Elekes-Szabó in the last decade. I will overview some of the recent developments in this area and present a new general theorem joint with Peterzil and Starchenko that provides uniform bounds in arbitrary (co-)dimension, in particular answering a question of Bays and Breuillard. Title: The AI Mathematician Abstract: We summarize how AI can approach mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and string theory, we present a number of recent experiments on how various standard machine-learning algorithms can help with pattern detection across disciplines ranging from algebraic geometry to representation theory, to combinatorics, and to number theory. At the heart of the programme is the question how does AI help with mathematical discovery. Title: Algebraic properties in families of linear differential equations Abstract: We study families of linear differential equations with rational coefficients, i.e., linear differential equations whose coefficients are rational functions of some parameters and the independent variable. If an algebraic property, such as not all solutions being Liouvillian functions, holds for generic parameters, i.e., if the parameters are considered to be algebraically independent, then it is natural to expect that the same algebraic property also holds for most values of the parameters. But what exactly do we mean by "most"? This is joint work with Ruyong Feng. Title: Morita equivalence for operator systems Abstract: In ring theory, Morita equivalence generalizes the isomorphism equivalence between commutative rings preserving several structural properties. A strong Morita equivalence for selfadjoint operator algebras was introduced by Rieffel in the 60s, and works as a correspondence between their representations. In the past 30 years there has been an interest to develop a similar theory for nonselfadjoint operator algebras and operator spaces with much success. Taking motivation from recent work of Connes and van Suijlekom, we will present a Morita theory for operator systems. We will give equivalent characterizations via Morita contexts, bihomomoprhisms, stable isomorphism and tensorial decompositions, while we will highlight properties that are preserved in this context. Time permitted we will provide applications to rigid systems, function systems and non-commutative graphs. This is joint work with George Eleftherakis and Ivan Todorov. Title: Frieze patterns: from combinatorics to representation theory Abstract: Friezes are infinite arrays of numbers, in which every four neighbouring vertices arranged in a diamond satisfy the same arithmetic rule. Introduced in the late 1960s by Coxeter, and further studied by Conway and Coxeter in their remarkable papers from 1973, this topic has been nearly forgotten for over thirty years. But recently, frieze patterns have attracted a lot of interest and appeared in many areas of mathematics, like combinatorics, geometry, and representation theory. In this talk I will review the beautiful Conway-Coxeter theorem relating Coxeter's frieze patterns to triangulations of polygons and then focus on recent developments in representation theory, in particular in connection with cluster algebras and cluster categories. Title: Structure and algorithms for even-hole-free graphs Abstract: The class of even-hole-free graphs (i.e. graphs that do not contain a chordless cycle of even length as an induced subgraph) has been studied since the 1990's, initially motivated by their structural similarity to perfect graphs. It is known for example that they can be decomposed by star cutsets and 2-joins into algorithmically well understood subclasses, which has led to, for example, their polynomial time recognition. Nevertheless, the complexity of a number of classical computational problems remains open for this class, such as the coloring and stable set problems. In this talk we survey some of the algorithmic techniques developed in the study of this class. Title: When is a Mathematical Object Well-Behaved? Abstract: In this talk we will come at this question from two different angles: first, from the viewpoint of model theory, a subject in which for nearly half a century the notion of stability has played a central role in describing tame behaviour; secondly, from the perspective of combinatorics, where so-called regularity decompositions have enjoyed a similar level of prominence in a range of finitary settings, with remarkable applications. In recent years, these two fundamental notions have been shown to interact in interesting ways. In particular, it has been shown that mathematical objects that are stable in the model-theoretic sense admit particularly well-behaved regularity decompositions. In this talk we will explore this fruitful interplay in the context of both finite graphs and subsets of abelian groups. To the extent that time permits, I will go on to describe recent joint work with Caroline Terry (The Ohio State University), in which we develop a higher-arity generalisation of stability that implies (and in some cases characterises) the existence of particularly pleasant higher-order regularity decompositions. Title: Introduction to mean curvature flow Abstract: A geometric flow is a method of deforming smooth geometric objects such as surfaces in 3 dimensions. They have been central to proving a wide variety of exciting theorems in differential geometry and topology. In this talk I will introduce one particular flow, mean curvature flow, and give some idea of the properties of such flows and their uses. There will be a lot of pictures and I will not assume any background in geometry. Title: Rouquier blocks for Ariki-Koike algebras Abstract: The Rouquier blocks, also known as the RoCK blocks, are important blocks of the symmetric groups algebras and the Hecke algebras of type A, with the partitions labelling the Specht modules that belong to these blocks having a particular abacus configuration. We generalise the definition of Rouquier blocks to the Ariki-Koike algebras, where the Specht modules are indexed by multipartitions, and explore the properties of these blocks. Title: Encoding arithmetic in Galois groups Abstract: Following the footsteps of Galois, we shall try to recover arithmetic properties from Galois theoretic data. In particular, we shall analyze what arithmetic information about a field K is encoded in its absolute Galois group G_K, i.e., the Galois group of a separable closure of K over K [this is the universal Galois group over K in the sense that every other Galois group over K is a quotient of G_K]. We shall see that for real closed or p-adically closed K, G_K knows everything about the arithmetic of K, and if K is the field of rational numbers, G_K knows almost everything - we shall make this precise in the talk, we shall relate it to the Section Conjecture in Grothendieck's so-called anabelian geometry and we shall describe model theoretic variants of these phenomena. Title: Maximal Subgroups in Special Inverse Monoids Abstract: I will give a gentle introduction to special inverse monoids, including the motivation for studying them, and discuss some recent joint work which Robert D. Gray on their subgroup structure. Title: Functional calculus in geometry and global analysis Abstract: Functional calculus, the ability to take a function of an operator, emerged in the latter half of last century as a convenient and conceptual tool particularly in the analysis of partial differential equations. In the last thirty years, methods from real-variable harmonic analysis have become of importance in establishing functional calculus. In the setting of geometry, functional calculus can be seen as an operator-dependent Fourier theory, able to capture the "global" aspect of geometric problems. The goal of this talk will be to flesh out a brief narrative of the journey of functional calculus, how it came to interact with harmonic analysis, and the way it has entered geometry in recent times. It will culminate with state-of-the-art results, but the beginnings will be humble, starting with the Fourier series! For the majority of the talk, no background will be assumed beyond a nodding acquaintance of Hilbert spaces, self-adjoint operators, and the spectrum of an operator. Title: What, how, and why, a story of forcing Abstract: Forcing is one of the key techniques in modern set theory. It is one of the main tools with which we study provability and independence. In this talk we will talk about what is set theory, how it came to be a foundation of mathematics (and what does that even mean?), as well as the basic ideas behind forcing and what it can and cannot do. All the way to some of the cutting-edge research that is done in room 8.32 on that subject. |